> Carsten Butz:
> > Would they agree that Brouwer or Bishop would have been able to
> > build a bridge?
>> Stephen Simpson:
> > I don't think this is the right question. A better question is: Would
> > Brouwer or Bishop have been able to build a bridge using
> > intuitionistic or constructive mathematics? There is a big difference
> > between these two questions, [...]
> >
> > I have my doubts about whether intuitionistic and/or constructive
> > mathematics is "enough to build bridges".
Ironically, Brouwer would have been on Simpson's side here. There is a
booklet he wrote, called "Life, Art, Mysticism" (it was written in Dutch,
is not in his Collected works because it contains political statements Heyting
didn't like; I don't know whether there is an English translation) in which
he says that the very idea that Mathematics can be applied at all to the real
world, is a form of hybris. There are probably also other papers where he
expressed this view.
One can doubt whether intuitionistic mathematics is sufficient for building
bridges, since although there are quite a few building blocks for ordinary
analysis available (suitable forms of Taylor's theorem, fundamental theorem
of calculus, Fourier and Laplace transform, solution of differential
equations), the treatment is so laborious that it was never completely
carried out (Bridges is still at it, though).
But, mathematics is not about building bridges; it is a continuous development
of ideas and new areas and in giving foundations for this (that is, a unified
picture), intuitionism does very badly. Consider the following example from
"almost everyday mathematics": the fundamental group of a (pointed) topological
space. Classically the picture is nice: loops in the space, formalized as
continuous maps from the unit interval into the space. But intuitionistically
these cannot be composed because we cannot define continuous functions making
case distinctions. This is definitely a problem for thorough intuitionists, and
probably the reason why there are so few left.
Jaap van Oosten
Dept. of Mathematics,
Utrecht University,
The Netherlands